In a more specific situation where \(W_{i}=(i-1)/n\)  are deterministic and \({\gamma }^{*}:[0,1)\rightarrow I\subset {\mathbb {R}}\)  is a right-continuous step function with an unknown number \(N-1\)  of changepoints (i.e. \(N\)  segments, \(N\ge 1\) ), the estimation problem we consider here immediately fits the setting of changepoint detection problem in exponential families. In this context, [1]} proposed a simultaneous multiscale changepoint estimator (SMUCE for short). More precisely, for each candidate estimator, [1]} designed a multiscale statistic to evaluate the maximum over the local likelihood ratio statistics on all discrete intervals such that the estimator is constant on these intervals with some value. Then provided a threshold \(q\) , the quantity \(N\)  is estimated by \(\widehat{N}(q)\)  which is the number of segments of the estimators satisfying their threshold condition with the minimal segments. Finally, their estimator is the likelihood maximizer over a constrained set in which all the estimators satisfy the threshold condition with exact \(\widehat{N}(q)\)  segments. [3]} ([3]}, [5]}) considered partitions given by the pruned dynamic programming algorithm ([6]}) and proposed a penalized log-likelihood estimator following the works of constructing the penalty function done by L. Birgé and P. Massart (see [7]} and [8]} for instance). They showed that their resulting estimator satisfies some oracle inequalities. One common feature of the above mentioned two methods is that they are both, more or less, based on the maximum likelihood estimation. When the data set contains a small amount of outliers, both of the two procedures infer extra changepoints to fit the outliers while identifying the true ones in the signal. For this point, we shall illustrate it in a more straightforward way in the simulation part of this paper. A natural question is can we find a procedure to enhance the stability of their estimators?
